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  1. bananabandana

    Thomson Scattering In the Early Universe

    Homework Statement Why does Thomson scattering occur in the early universe? Homework Equations $$ e^{-} + \gamma \rightarrow e^{-} + \gamma $$ is a Thomson scattering process if: $$ E_{\gamma} << m_{e}c^{2}$$ (Electrons are essenitally stationary) The Attempt at a Solution [/B] Very...
  2. bananabandana

    Transforming to a Rotating Reference Frame

    Homework Statement Let ## \mathbf{r} ## be the position of a point in a rigid body relative to some origin ##O##. Let ##\mathbf{R}## be the position of the centre of mass from that origin. ##\mathbf{r^{*}} = (\mathbf{r}-\mathbf{R})##. ## d\boldsymbol{\phi} ## is the infitesimal vector directed...
  3. bananabandana

    Link between 'time' component of 4-momentum and energy

    Sure, sure - the units are important and I entirely agree that ## p^{0}c ## has the correct units of energy. What I find confusing is not the units, but the idea that somehow a translation in time is related to my classical picture of energy :)
  4. bananabandana

    Link between 'time' component of 4-momentum and energy

    Good question - I presume it's the four velocity of the observer (these are directly from my lecture notes). I guess another part of my confusion is the fact that we have : $$ p^{\mu} = m \frac{dx^{\mu}}{d\tau} \\[2mm] p^{\mu} = \bigg(E,\vec{\dot{x}} \bigg) \\[2mm] \implies...
  5. bananabandana

    Link between 'time' component of 4-momentum and energy

    Homework Statement $$ E = -\vec{v_{obs}} \cdot \vec{p} $$ Where ## \vec{p} ## is the four momentum, and ## \vec{v_{obs}}## the velocity of the observer. Homework Equations The Attempt at a Solution [/B] This was a stated result in a GR course. I look through my SR notes and find that I...
  6. bananabandana

    General proof of Arc Length For Parametrised Coodrdinates

    It's not generally true that the off diagonal elements of the metric tensor are zero - we require that the tensor be symmetric, but that's not *necessarily* the same as it being diagonal? Even if I've got that wrong though...If you think of ##S^{2}##, then the metric definitely isn't just...
  7. bananabandana

    General proof of Arc Length For Parametrised Coodrdinates

    Homework Statement Prove that, given a metric ##g_{ij}## such that ##ds^{2}=g_{ij}dx^{i}dx^{j}##, where ##x^{r} = x^{r}(\lambda)## , we have the following result for the arc length: $$ L(p,q) = \int_{p}^{q} ds = \sqrt{ g_{ij} \frac{dx^{i}}{d \lambda} \frac{ dx^{j}}{d \lambda} } d \lambda $$...
  8. bananabandana

    Percolation Problem

    Homework Statement We have a 1-D lattice [a line] of ##L## sites. Sites are occupied with probability ##p##. Find the probability that a given site is a member of a cluster of size ##s##. (A cluster is a set of adjacent occupied sites. The cluster size is the number of occupied sites in the...
  9. bananabandana

    Superposition of Two Travelling Waves, Different Amplitudes

    Ah, that's embarrassing. Out of interest - why is the cosine rule idea not valid? Please see picture. **EDIT**: Oops, in the diagram ##\vec{A}## and ##\vec{C}## have the opposite phase to as written in the question.
  10. bananabandana

    Superposition of Two Travelling Waves, Different Amplitudes

    Homework Statement I'm looking at an E&M textbook - "Time-Harmonic Electromagnetic Fields". They state: "A more general ##x ## polarized field is one consisting of waves traveling in opposite directions with unequal amplitudes - i.e : (1) $$ E_{x} = Ae^{-jkz} +Ce^{jkz}$$ Let ## A ## and...
  11. bananabandana

    Calculating Flux through Ellipsoid

    So long as it's set up right, I'm happy. Thanks for the help! :)
  12. bananabandana

    Calculating Flux through Ellipsoid

    Oh, of course, because the diagonal of the square is ## \sqrt{2} ##. I see! Thanks. I think I am leaving ## z ## in the ##z## column though, but the vector field ## \vec{F} ## is ##\vec{F}=(-y,x,0) ## - or did you mean something else? But am I correct in thinking now that the answer I first...
  13. bananabandana

    Calculating Flux through Ellipsoid

    Ah, sorry. I forgot to say that the question stated ## a > \sqrt{2} ## , ## b >\sqrt{2} ## and ## a \neq b ##. Might this change things? (Though I admit, I don't know how they got these conditions).
  14. bananabandana

    Calculating Flux through Ellipsoid

    Isn't the region of integration a square? ( I thought the definition of the region as ## 0 \leq x \leq 1 ## and ## 0 \leq y \leq 1 ## implied this?). Hence, no need for writing ## y=y(x)## in the limits... and yes, I know I can get it from (2), but the question was phrased in a way that it...
  15. bananabandana

    Calculating Flux through Ellipsoid

    Homework Statement Let ## E ## be the ellipsoid: $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$ Let ## S ## be the part of the surface of ## E ## defined by: $$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$ Let F be the vector field defined by $$ F=(-y,x,0)$$ A) Explain why ##...
  16. bananabandana

    Derivation of the Continuity Equation for Fluids

    Ah, no that's not at all what I meant. Sorry. I must have said something confusing :P When I first wrote the above down, I was imagining the density as just a constant. Looking back I see no problem if ## \rho = \rho(x,y,z)## since ## dm## is infinitesimal. (though $\rho$ must be independent...
  17. bananabandana

    Derivation of the Continuity Equation for Fluids

    The component of the fluid velocity flowing through the small surface element ## dA ##. ## \rho ## may be a function of position and time, but why would this make the above invalid? Since we're using an infitesimal mass, ## \rho ## might (almost) as well be a constant, no? i.e. we're...
  18. bananabandana

    Derivation of the Continuity Equation for Fluids

    Homework Statement Derive a mathematical relationship which encapsulates the principle of continuity in fluid flow. Homework Equations The Attempt at a Solution Imagine we have a mass of fluid ## M##, of volume ##V##, bounded by a surface ##S##. If we take a small element of this volume...
  19. bananabandana

    Inverse Jacobian

    Homework Statement Don't understand why the inverse jacobian has the form that it does. Homework Equations $$ J = \begin{pmatrix} \frac{\partial{x}}{\partial{u}} & \frac{\partial{y}}{\partial{u}} \\ \frac{\partial{x}}{\partial{v}} & \frac{\partial{y}}{\partial{v}} \end{pmatrix} $$ $$...
  20. bananabandana

    Two variable function, single integral

    Thanks LCKurtz t=tan(\frac{x}{2}) works. Had completely forgotten about that, silly me :P
  21. bananabandana

    Two variable function, single integral

    Homework Statement Evaluate: I(y)= \int^{\frac{\pi}{2}}_{0} \frac{1}{y+cos(x)} \ dx if y > 1 Homework Equations The Attempt at a Solution I've never seen an integral like this before. I can see it has the form: \int^{a}_{b} f(x,y) dx I clearly can't treat it as one half of an exact...
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